Method for Determining the Respective Temperature of Several Battery Cells of a Vehicle Battery Via Extrapolation of a Measured Temperature; Control Device and Vehicle Battery

ABSTRACT

A method for determining respective temperatures of a plurality of battery cells includes determination of measured values including a first voltage of a first battery cell, a second voltage of a second battery cell, and a respective current that flows through the first and second battery cells. A first measured resistance of the first battery cell and a second measured resistance of the second battery cell is determined from the measured values. A reference resistance is determined. A first resistance relationship from the first measured resistance and the reference resistance and a second resistance relationship from the second measured resistance and the reference resistance are determined. A measured temperature of the first battery cell is determined. A computed temperature of the second battery cell is determined based on the measured temperature of the first battery cell, the first resistance relationship, and the second resistance relationship according to a predetermined requirement.

BACKGROUND AND SUMMARY OF THE INVENTION

The invention relates to a method for determining the respectivetemperature of several battery cells of a vehicle battery. A secondaspect of the invention relates to a corresponding control device. Athird aspect of the invention relates to a vehicle battery having thestated control device, among others.

Electric motor vehicles must be able to fulfil the high expectations andrequirements that are already fulfilled by conventional fuel-poweredvehicles with regards to lifespan, performance, range and safety for asuccessful commercialisation. Purely electric vehicles that only have anelectrical energy storage and one or several electric motors, hybridvehicles that have a combustion engine, an electrical energy storage andone or several electric motors, and hydrogen vehicles that, for example,have a fuel cell, an electrical energy storage and one or severalelectric motors, are examples of electric motor vehicles. An electricalenergy storage of a motor vehicle can also be described as a vehiclebattery and/or traction battery. A vehicle battery is composed of aplurality of battery cells that are interconnected among each other inseries and/or in parallel. The plurality of battery cells is designed tosave electrical energy collectively, or to collectively provide anelectrical power. The battery cells of a vehicle battery can all beinterconnected in series. It is advantageously provided, however, thatseveral battery cells are interconnected with each other in series, andyield a group wherein several such groups are interconnected inparallel. The nominal voltage level of the vehicle battery results fromthe nominal voltage of the individual battery cells and the number ofbattery cells interconnected in series. The nominal voltage level ispreferably at least 80 volts, preferably at least 200 volts, for example400 volts or 800 volts.

It is in particular provided that the vehicle battery is continuouslymonitored in order to guarantee an efficient and safe operation of thevehicle battery or of the motor vehicle having the vehicle battery.Within the context of the monitoring, the respective temperature,voltage, current, state of charge (SoC) and/or the degree of wear ordegradation (state of health, SoH) is checked, for example. Anycombination of the variables given as examples is here possible. It isextremely advantageous to continuously check, or to determine and tomonitor the respective temperature of each of the battery cells in anoperation. This applies analogously to the degree of wear or thedegradation (SoH).

The arrangement of a respective temperature sensor on each battery cellis here extremely complicated and cost intensive. Correlations betweenthe given variables and the internal resistance of the respectivebattery cells are frequently utilised for this reason. The temperatureand/or the degradation of the respective battery cell can in particularbe determined from the internal resistance.

The internal resistance of a respective battery cell can for example beused by means of a current pulse measuring method that utilises acorrelation of a voltage response to a current pulse. A lowcomputational effort is necessary for this purpose. Values determined inthis way can only be provided irregularly, however. An alternativemethod uses an adaptive model, for example a so-called electric circuitmodel (ECM). This relates in particular to a Kalman filter.

It is the object of the present invention to enable an improvedmonitoring of the temperature of respective battery cells of a vehiclebattery.

A first aspect of the invention relates to the method for determiningthe respective temperature of several battery cells of a vehiclebattery. In order to enable the improved monitoring of the respectivetemperature, the method has the following steps:

-   determination of measured values, comprising a first voltage of at    least one first battery cell and a second voltage of at least one    second battery cell and at least one respective current that flows    through the battery cells,-   determination of a first measured resistance of the first battery    cell and a second measured resistance of the second battery cell    from the measured values,-   determination of a reference resistance,-   determination of a first resistance relationship from the first    measured resistance and the reference resistance and of a second    resistance relationship from the second measured resistance and the    reference resistance,-   determination of a measured temperature of the first battery cell,-   determination of a computed temperature of the second battery cell    by means of the measured temperature and the first resistance    relationship and the second resistance relationship according to a    predetermined requirement.

The determination of the measured values occurs in particular via acorresponding measurement by means of a measurement device. Themeasurement device can in particular be designed to measure the firstvoltage and/or the second voltage and/or the respective current. Therespective current that flows through the first battery cell and thesecond battery cell is in particular the same current, as both batterycells are interconnected in series. The first measured resistance andthe second measured resistance can be determined by means of themeasured values. The determination of the first and second measuredresistance occurs in particular by means of a mathematical correlationbetween current voltage and resistance. The determination of the firstmeasured resistance of the first battery cell and the second measuredresistance of the second battery cell from the measured values occurs bymeans of Ohm’s law, for example. More complicated mathematicalcorrelations can also be used in a further embodiment, as will furtherbe shown in the further course of this application.

The reference resistance can be a predetermined value, for example. Thedetermination of the reference resistance can occur in this case byrequesting the predetermined value from a storage device. The referenceresistance can alternatively be formed as an average value of allmeasured resistances of the several battery cells, for example the firstmeasured resistance and the second measured resistance.

The determination of the reference resistance occurs in particular bymeans of averaging or forming the average of all measured resistances ofall battery cells of the vehicle battery. The first and the secondmeasured resistance relationship can be formed by dividing therespective measured resistance, and thus the first or the secondmeasured resistance, by the reference resistance. The first or secondreference resistance can alternatively be formed by distributing thereference resistance by the first or the second reference resistance.

The determination of the measured temperature of the first battery cellcan occur by measuring the temperature of the first battery cell. Thedetermination or measurement of the measured temperature in particularoccurs by means of a temperature sensor. The temperature sensor can forexample be formed as an NTC or PTC sensor, or comprise a sensor of thiskind. A measurement by means of an infrared thermometer is alternativelyalso possible. Any other measuring method for measuring a temperature ofthe first battery cell is also conceivable. The measured temperature ofthe first battery cell can generally be determined by means of atemperature measuring unit.

The determination of the computed temperature occurs on the basis of thefirst resistance relationship und the second resistance relationship,and the measured temperature according to the predetermined requirement.The computed temperature here indicates the temperature of the secondbattery cell, while the measured temperature indicates the measuredtemperature of the first battery cell. The computed temperature isdescribed as such as it is not measured, but is rather derived ordetermined from the given variables by means of the predeterminedrequirement.

The invention is based on the idea of extrapolating the temperature ofthe second battery cell from the temperature of the first battery cell.The measured temperature is compared with the first and the secondresistance relationship for this extrapolation. The invention here takesadvantage of the fact that the internal resistance of a respectivebattery cell decreases when the temperature of the corresponding batterycell increases. The temperature of the first battery cell can thus beextrapolated to the second battery cell based on the respective internalresistance, and thus on the first and second measured resistance of thefirst and second battery cell. Here it is in particular taken intoaccount that there is no linear relationship between the temperature andthe respective internal resistance. The relationship between respectiveinternal resistance and temperature can be available via a distributionfunction and/or a corresponding value table, for example. Therelationship, in particular the distribution function or the valuetable, can here be predetermined. The relationship, in particular thedistribution function or the value table, is in particular derived viacorresponding tests on a vehicle battery of this kind or correspondingbattery cells. A numerical generation of the value table or thedistribution function can here be provided. Overall, it is thus shownthat an improved monitoring of the temperatures of respective batterycells of a vehicle battery can be enabled. In particular, only a lownumber of temperature measuring units or temperature sensors can besufficient to determine the respective temperature of all battery cellsvia the method according to the invention. This saves complexity in theconstruction of the vehicle battery on the one hand, and costs on theother.

The method is generally also possible without the steps of determiningthe reference resistance and determining the first and second resistancerelationship. In this case, the computed temperature could occurdifferently by means of the measured temperature, and the first measuredresistance and the second measured resistance according to acorresponding predetermined requirement. Here it has proved to be thecase, however, that the comparison with the reference resistance, alsodescribed as normalization, can deliver reliable results. This here inparticular proceeds from the knowledge that the absolute resistances arenot of interest, as the respective battery cell having the lowestinternal resistance is usually also the warmest of the battery cells,and the battery cell having the highest internal resistance is usuallythe coolest of the battery cells. As the determination of the computedtemperature also occurs independently of the measured temperature, theobservation of the respective measured resistance normalized to thereference resistance, and thus to the first or second resistancerelationship, is sufficient. A homogeneous database is obtained vianormalization.

It is provided according to a development that the predeterminedrequirement contains the distribution function and/or the value tablethat assign a respective value, in particular exactly one respectivevalue, for the temperature, and thus the computed temperature, to aplurality of values for the second resistance relationship, depending onthe measured temperature and the first resistance relationship. It canhere be provided that the distribution function and/or the value tableare generated via the corresponding tests or measurements on the vehiclebattery. The value table or the distribution function can be providedvia a mathematical relationship or a mathematical formula, ornumerically. The value table and/or the distribution function canadditionally have the discharge state of the vehicle battery and/or therespective battery cell as parameters. In other words, the respectiverelevant distribution function and/or value table can be consulted fordifferent values of the state of charge of the vehicle battery and/orthe respective battery cell. In this case, the method can contain thedetermination of a state of charge of the vehicle battery and/or therespective battery cell as an additional step. An even more precisedetermination of the temperature, in particular computed temperature ofthe second battery cell, can be obtained in this way.

It is provided according to a development that the distribution functionand/or value table is first normalized depending on the first resistancerelationship and the measured temperature, and the computed temperatureis then derived from the normalized distribution function and/or valuetable depending on the second resistance relationship. In other words,first the distribution function and/or value table is selected orgenerated by means of the data of the first battery cell, and thus thefirst resistance relationship and the measured temperature, and then thecomputed temperature is selected from the distribution function and/orvalue table generated in this way, depending on the second resistancerelationship. The normalized distribution function and/or value tablecan here indicate a one-to-one relationship between computed temperatureand second resistance relationship. In this way, it is possible toeasily derive the computed temperature by means of the normalizeddistribution function and/or value table. The normalization orgeneration or selection of the normalized distribution function and/orvalue table can occur in particular on the basis of predetermined data.

It is provided according to a development that a respective secondresistance relationship for several second battery cells is determined,from which, and by means of the predetermined requirement, the measuredtemperature and the first resistance relationship, a respective valuefor the computed temperature of the respective second battery cell isdetermined. In other words, the determination of the computedtemperature of the second battery cell occurs in an analogue manner forseveral second battery cells in parallel. The respective secondresistance relationship can here be used as the basis for each of thesecond battery cells, the second resistance relationship being derivedfrom a respective measured resistance that is determined for therespective second battery cell. A respective second voltage can bedetermined for each of the second battery cells to determine therespective second measured resistance. A respective current can bedetermined for each of the second battery cells, wherein the respectivecurrent can be the same for all second battery cells in the event of aninterconnection in series. The respective computed temperature of therespective second battery cell is here determined in each case on thebasis of the first resistance relationship of the first battery cell andthe measured temperature of the first battery cell. In other words, thecomputed temperature of each of the second battery cells is extrapolatedon the basis of the measured temperature of the first battery cell andits resistance or resistance relationship. In this way, a plurality oftemperature sensors is not required, as a temperature sensor is nolonger needed for each of the second battery cells.

It is provided according to a development that a respective measuredtemperature and a respective first resistance relationship is determinedfor several first battery cells, from which, and by means of thepredetermined requirement and the second resistance relationship, arespective value for the computed temperature of the second battery cellis determined. In other words, the computed temperature of the secondbattery cell is extrapolated on the basis of the respective measuredtemperature and the respective first resistance relationship of severalfirst battery cells. The determination or extrapolation of the computedtemperature occurs in particular on the basis of the different batterycells independently of each other. In a further embodiment, it can beprovided that the several values for the computed temperature of thesecond battery cell are averaged. Outliers, and thus values for thecomputed temperature of the second battery cell that deviate from theusual values by more than a predetermined amount, can be filtered outbefore this averaging occurs. This is because values deviatingsignificantly in this way or outliers are probably due to errors in themeasuring or errors in the determination of the computed temperature. Inthis way, the precision in the determination of the respectivetemperature of the battery cells can be further improved.

It is provided according to a development that at least one measuredtemperature of one of the several first battery cells arranged on theedge of the vehicle battery and at least one measured temperature of oneof the several first battery cells arranged in the center of the vehiclebattery is determined. In other words, a respective temperature sensoror a respective temperature measuring unit is arranged on at least onefirst battery cell that is arranged on the edge of the vehicle battery,and at least one temperature sensor or temperature measuring unit isarranged on a first battery cell that is arranged in the middle of thevehicle battery. The first battery cell arranged on the edge of thevehicle battery is only next to one further battery cell, for example.The first battery cell arranged in the middle of the vehicle battery isnext to a respective further battery cell on several sides, for example.In this way, different temperature measured values for differentinstallation situations of the different first battery cells result.This enables an even more precise determination of the measuredtemperature, and thus also of the computed temperature or the severalcomputed temperatures derived from it.

It is provided according to a development that it is checked whether areference state is present, wherein a correlation coefficient G isgreater than a predetermined value and/or respective measuredtemperatures of several first battery cells among themselves do notexceed a predetermined measurement in the reference state. Thecorrelation coefficient G applies in particular to the quality of aregression of the determination of the respective measured resistances.The correlation coefficient G here represents in particular a qualityfactor that characterises the data quality of the measured values orresistance values derived from them. The reference state is here inparticular a static state in which the vehicle battery is in a staticstate. The determination of the respective temperature of the severalbattery cells is not required in the reference state, or exactly whenthe reference state is present, as it can be assumed that these developin a temporally constant manner. In addition, other evaluations can bemade exactly when the reference state is present. A quantification ofthe degradation or a change of the degradation of the vehicle battery incomparison with earlier reference states occurs in particular. Thisproceeds from the knowledge that slow or long-term behaviour of themeasured value is not caused by a change of the temperature of thebattery cells, but by a change of their degradation. The change of thestate of the vehicle battery or individual cells or their degradationcan thus be determined from this long-term behaviour.

A second aspect of the invention relates to a control device fordetermining the respective temperature of several battery cells of avehicle battery, wherein the control device is designed to:

-   receive measured values, comprising a first voltage of at least one    first battery cell and second voltage of at least one second battery    cell and at least one respective current that flows through the    battery cells,-   determine a first measured resistance of the first battery cell and    a second measured resistance of the second battery cell from the    measured values,-   determine a reference resistance,-   determine a first resistance relationship from the first measured    resistance and the reference resistance, and a second resistance    relationship from the second measured resistance and the reference    resistance,-   receive a measured temperature of the first battery cell, and-   determine a computed temperature of the second battery cell by means    of the measured temperature and the first resistance relationship    and the second resistance relationship according to a predetermined    requirement.

The control device is in particular designed to carry out the methodaccording to the invention according to one or several of theembodiments described here. The control device comprises a computingunit, for example, the computing unit being formed as a microcontroller,field programmable gate array (FPGA) or digital signal processor (DSP).The control device or the computing unit can have a memory unit, forexample a flash memory, a magnetic memory medium, and/or an opticalmemory medium or the like, in which a computer program product is savedthat contains program coding means to carry out the method according tothe invention or individual method steps of the method according to theinvention. The program coding means in particular provide for theexecution of the method according to the invention according to one orseveral embodiments when executed on the control device or the computingdevice.

A third aspect of the invention relates to the vehicle battery having:

-   the control device mentioned above,-   at least one first battery cell and a respective temperature    measuring unit for determining the respective measured temperature    of each first battery cell,-   at least one second battery cell, and-   a measuring device for determining measured values, [comprising] a    first voltage of a first battery cell and second voltage of a second    battery cell, and at least one respective current that flows through    the battery cells.

The vehicle battery can be executed as a lithium-ion battery, forexample. The vehicle battery can have a voltage level of more than 80volts, preferably more than 200 volts, for example 400 volts or 800volts. The vehicle battery in particular has several temperaturemeasuring units, for example two, three or four. In this case, fourbattery cells of the vehicle battery are first battery cells in thesense of the present application. The respective temperature of theremaining battery cells of the vehicle battery, the battery cells alsobeing able to be described as second battery cells in the sense of thepresent application, is extrapolated by means of the respective measuredtemperatures of the first battery cells or the one first battery cell.

Further advantages, features and details of the invention result fromthe following description of preferred exemplary embodiments and withreference to the drawings. The features and combinations of featurespreviously given in the description, and the features and combinationsof features given in the following description of figures and/or shownsolely in the figures can be used not only in the respectively givencombination, but also in other combinations or in isolation withoutleaving the scope of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a schematic block diagram of a vehicle battery havingseveral battery cells, wherein some of the battery cells have arespective temperature sensor, and the temperature of the remainingbattery cells is extrapolated by means of the measured temperatures andrespective resistances of the battery cells;

FIG. 2 shows an exemplary method sequence with reference to a processdiagram; and

FIG. 3 shows an exemplary experimentally provided distribution function.

DETAILED DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a vehicle battery 1 having a plurality of battery cells 2according to an extremely exemplary embodiment. The battery cells 2 aredivided into two groups 3, wherein the battery cells 2 of a respectivegroup 3 are respectively interconnected in series. The groups 3 areinterconnected with each other in parallel in turn. Every otherreasonable interconnection of the battery cells 2 is of course alsopossible. The vehicle battery 1 is in particular formed as a so-calledtraction battery to provide an electric drive of a motor vehicle havingelectrical energy to drive or accelerate a motor vehicle. The vehiclebattery 1 or its battery cells 2 are in particular based on thetechnology of a lithium-ion battery.

The vehicle battery 1 has electrical ports 5 to which an output voltageof the vehicle battery 1 is applied. A load, in particular an inverterfor operating an electric machine of a motor vehicle and/or a voltagetransformer for providing an on-board power supply of the motor vehiclewith electrical energy can in particular be applied to the electricalports 5. In other words, the vehicle battery 1 can emit electricalenergy or an electrical power outwards via the electrical ports 5. Theelectrical ports 5 for providing the electrical power can be the onlypower requirements that are led out of a battery housing 6 of thevehicle battery 1. The electrical ports 5 can be provided by two polesin total. The output voltage of the vehicle battery 1 is nominally inparticular more than 80 V, preferably more than 200 V, for example 400 Vor 800 V. The number of battery cells 2 interconnected in series alsoderives from the desired nominal voltage level of the output voltagewhen lithium-ion based battery cells 2 are used.

A monitoring of the vehicle battery 1 that is as comprehensive aspossible is necessary in order to be able to guarantee a high degree ofsafety during the operation of the vehicle battery 1 and to maximise alifespan of the vehicle battery 1. The monitoring can for example relateto the output voltage, a current flow, a temperature, a state of charge(SOC), a degradation (SOH) or the like. It is particularly preferred ifthis monitoring occurs not only in relation to the entire vehiclebattery 1, but at least partially for each of the battery cells 2individually. In this way, damage to individual battery cells 2 can berecognised or prevented.

The vehicle battery 1 has several temperature measuring units 4. Thetemperature measuring units 4 are assigned to a respective battery cell2. The temperature measuring units 4 are here designed to measure thetemperature of the respective battery cells 2. The battery cells 2 towhich a temperature measuring unit 4 is assigned are also described asfirst battery cells 11. It is obvious that a temperature measuring unit4 is only assigned to a fraction of the battery cells 2. The batterycells 2 to which no temperature measuring unit 4 is assigned are alsodescribed as second battery cells 12. Only a few temperature measuringunits 4 are provided, in order to keep a complexity and production costsof a vehicle battery 1 as low as possible. On the other hand, it isdesirable to determine the respective temperature of all the batterycells 2. For this reason, the measured temperature of the first batterycells 11 is extrapolated to the second battery cells via a method fordetermining the respective temperature of battery cells of the vehiclebattery 1. The vehicle battery 1 can have a corresponding control device9 to carry out this method.

The vehicle battery 1 presently has a measuring device 7 that isdesigned to measure or determine current and voltage of the individualbattery cells 2. Here it can in particular be provided that a respectivevoltage for each of the battery cells 2 is measured or determined. Itcan be provided that the current for each of the battery cells 2 ismeasured independently of each other. The current is preferably measuredor determined for several battery cells 2 together, however. Therespective current that flows through the battery cells 2 interconnectedin series, presently the battery cells 2 of a group 3, is respectivelythe same due to the series interconnection of several battery cells 2.In this way, the current can respectively only be measured once for eachof the groups 3. The entire current of all the groups 3 canalternatively be measured and distributed over the number of the groups3, this occurs in particular under the assumption that the entirecurrent is equally distributed over all the groups 3. According to afurther alternative, the current of a group 3 can be measured and it canbe assumed that the current of the other groups 3 corresponds to thiscurrent. The measurement of the given measured value corresponds to astep S1 in relation to the process diagram according to FIG. 2 . Themeasured values are alternatively or additionally received via thecontrol device 9 in the step S1.

A respective resistance, also a measured resistance, can be determinedfor each of the battery cells 2 from the given measured values, and thusthe respective values for current and voltage. This can occur based onOhm’s law, for example. This occurs in other exemplary embodimentsaccording to different formalistic approaches that are describedsubsequently in more detail. This determination of the respectiveresistances or internal resistances of the individual battery cells 2corresponds to a step S2.

It is determined in a step S3 whether a reference state or a so-called“steady state” is present. The respective temperature of the firstbattery cells 11 differs from one another at most by a predeterminedamount in the present embodiment in this reference state. A correlationcoefficient G is also greater than a predetermined value in the presentembodiment in the reference state. The correlation coefficient G isdescribed in more detail in the following (see also formula (22)). If areference state of this kind is present, the method for determining therespective temperature is halted or carried out in a different manner.In this case, the temperature of the second battery cells 12 is notdetermined, this corresponds to the path “n” in the process diagram, butrather the degradation (“state of health”) of the battery is determined,which corresponds to the path “y” in FIG. 2 .

In a further step S6, the temperature values are requested or receivedfrom the temperature measuring units 4. This occurs in particular viathe control device 9. The measurement of the temperature values can alsoalternatively or additionally occur in the step S6 via the temperaturemeasuring units 4. The step S6 can be carried out repeatedly and/or becarried out before the step S3, such that the temperature values for thestep S3 are respectively provided in current form.

If no reference state is present, then the method can be resumed with astep S4. In a step S4, a reference resistance is determined. Thisreference resistance can for example be a predetermined value that isrequested during the step S4 from a memory unit of the control device 9.In the present exemplary embodiment, the reference resistance isdetermined from the internal resistances or measured resistances of allthe battery cells 2. The reference resistance is in particulardetermined via averaging of the same. In other words, the referenceresistance can be the average of the internal resistances or measuredresistances.

A respective resistance relationship is determined for each of theinternal resistances of the battery cells 2 in the step S5. Thisrespective resistance relationship is calculated by dividing therespective internal resistance or measured resistance by the referenceresistance. The respective resistance relationship can alternatively becalculated by dividing the reference resistance by the respectiveinternal resistance or measured resistance.

The actual extrapolation of the measured temperatures or temperaturevalues in relation to the first battery cells 2 occurs in the methodstep S7. A respective computed temperature of all the second batterycells 12 is here determined on the basis of the measured temperaturesvia the respective resistance relationships. The basis of the computedtemperature can here be a predetermined requirement. The predeterminedrequirement in particular connects the respective temperatures of thebattery cells 2 to the respective resistance relationship. Thepredetermined requirement can here contain a value table and/or adistribution function 15. The value table and/or distribution function15 can be derived in particular from corresponding laboratory tests. Theresults of the corresponding laboratory tests can be numericallycontained in the value table and/or distribution function, or can beemulated via a mathematical function.

FIG. 3 shows an exemplary distribution function 15 that specifies thetemperature T_(R) of a respective second battery cell 12, depending onthe respective resistance relationship V_(R2) of the correspondingsecond battery cells 12 and the state of charge SOC. The exemplarydistribution function 15 is here generated as a parameter on the basisof the first resistance relationship and the measured temperature of arespective first battery cell 11. In other words, a distributionfunction 15 based on the measured temperature and a first resistancerelationship of a respective first battery cell 11 is generated, inorder to derive the temperature T_(R) of a respective second batterycell 12 from the distribution function, depending on its resistancerelationship V_(R2) and state of charge SOC. In a further embodiment,this can occur for each second battery cell 12 independently of eachother on the basis of each first battery cell 11. In this way, as manycomputed temperatures are obtained for each battery cell 12 as firstbattery cells 11 or temperature measuring units 4 are present. Anaveraging of these several computed temperatures of each battery cell 12can occur. In a further embodiment, a sorting out of outliers orsignificantly deviating values can occur before this averaging. Aparticularly precise determination of the computed temperature resultsin this way.

The mathematical basis of the present invention will now be described inmore detail in the following.

A. Algorithm to Determine the Resistance

The basis for the determination or calculation of the respectiveinternal resistance is a sub-division of the course of the measuredvariables (current and voltage) into an instant system responseI(t)•R_(Ohm)(t) and a delayed system response F(t).

$\begin{matrix}{U(t) = I(t) \cdot R_{\text{Ohm}}(t) + F(t).} & \text{­­­(1)}\end{matrix}$

The instant system response is described via the voltage on the ohmicresistance R_(Ohm)(t) due to the current I(t). The delayed systemresponse F(t) is in turn sub- divided into a deterministic ratioF_(deter) and a stochastic noise ratio Fstoch

$\begin{matrix}{F(t) = F_{\text{deter}}(t) + F_{\text{stoch}}(t)} & \text{­­­(2)}\end{matrix}$

The delayed deterministic cell behaviour can be expressed by using anelectric circuit model (ECM) having two resistance pairs.

F_(deter)(t) = OCV(t) + U_(PoI)(t) + U_(Diff)(t),

$\frac{\text{d}U_{\text{PoI}}(t)}{\text{d}t} = - \frac{1}{R_{\text{PoI}} \cdot C_{\text{PoI}}} \cdot U_{\text{PoI}}(t) + \frac{I(t)}{C_{\text{PoI}}},$

$\frac{\text{d}U_{\text{Diff}}(t)}{\text{d}t} = - \frac{1}{R_{\text{Diff}} \cdot C_{\text{Diff}}} \cdot U_{\text{Diff}}(t) + \frac{I(t)}{C_{\text{Diff}}},$

$\frac{\text{dSoC}(t)}{\text{d}t} = \frac{I(t)}{C_{N}},$

The stochastic noise can be expressed via an independent Gaussian noisemethod having a mean average value of zero.

$\begin{matrix}{F_{\text{stoch}}(t) = N\left( {0,\sigma_{\text{noise}}^{2}} \right)} & \text{­­­(7)}\end{matrix}$

The values of the ECM parameters R_(Ohm), R_(Pol), R_(Diff), C_(Pol),C_(Diff) and C_(N) change over time depending on operating conditionsand cell states. The requirements of the usage of stochastic methodsrequire the voltage and current signals to be discretised into samplevalues. The sample size n and the sample rate T_(s) must fulfil tworequirements. On the one hand, the sampling size n must be large enoughto fulfil the stochastic measures with a high level of confidence,while, on the other hand, the ohmic resistance R_(Ohm)(t) and thedeterministic system behaviour in the temporal interval of the samplemust not be changeable Δt = n - T_(S).

On the basis of these requirements, the entire difference of the ohmicresistance

$\begin{matrix}\begin{matrix}{\text{Δ}R_{\text{Ohm}}(t) = \left( \frac{\partial R_{\text{Ohm}}}{\partial t} \right|_{T_{S}}\left( {\text{Δ}t + \frac{\partial R_{\text{Ohm}}}{\partial T} \cdot \frac{\partial T}{\partial t}} \right|_{T_{S}}\text{Δ}t + \ldots} \\{\left( {\ldots\frac{\partial R_{\text{Ohm}}}{\partial SoC} \cdot \frac{\partial\text{SoC}}{\partial t}} \right|_{T_{S}}\left( {\text{Δ}t + \frac{\partial R_{\text{Ohm}}}{\partial I} \cdot \frac{\partial I}{\partial t}} \right|_{T_{S}}\text{Δ}t \approx 0,}\end{matrix} & \text{­­­(8)}\end{matrix}$

must be negligibly small in the sample interval Δt. The entiredifference of the delayed system reaction leads to the followingequations

$\begin{matrix}{\text{Δ}F(t) = \text{Δ}\text{OCV}(t) + U_{\text{PoI}}(t) + \text{Δ}U_{\text{Diff}}(t) + \text{Δ}F_{\text{stoch}}} & \text{­­­(9)}\end{matrix}$

Under the condition that the alteration of the OCV

$\begin{matrix}\begin{matrix}{\text{Δ}\text{OCV}(t) = \left( \frac{\partial\text{OCV}}{\partial t} \right|_{T_{S}}\left( {\text{Δ}t + \frac{\partial\text{OCV}}{\partial T} \cdot \frac{\partial T}{\partial t}} \right|_{T_{S}}\text{Δ}t + \ldots} \\{\left( {\cdots + \frac{\partial\text{OCV}}{\partial\text{SoC}} \cdot \frac{\partial\text{SoC}}{\partial t}} \right|_{T_{S}}\text{Δ}t + \cdots \approx 0,}\end{matrix} & \text{­­­(10)}\end{matrix}$

is negated at relevant intervals. Due to the high temporal constants ofthe diffusion process

$\begin{matrix}{T_{S} < < R_{\text{Diff}} \cdot C_{\text{Diff}}} & \text{­­­(11)}\end{matrix}$

the second RC pair

$\begin{matrix}\begin{array}{r}{\text{Δ}U_{\text{Diff}}(t) = \left( \frac{\partial U_{\text{Diff}}}{\partial t} \right|_{T_{S}}\left( {\text{Δ}t + \frac{\partial U_{\text{Diff}}}{\partial T} \cdot \frac{\partial T}{\partial t}} \right|_{T_{S}}\text{Δ}t + \ldots} \\{\left( {\cdots + \frac{\partial U_{\text{Diff}}}{\partial\text{SoC}} \cdot \frac{\partial\text{SoC}}{\partial t}} \right|_{T_{S}}\text{Δ}t + \cdots \approx 0,}\end{array} & \text{­­­(12)}\end{matrix}$

can also be negated. Due to the low temporal constant, the entire changeof the first RC pair

$\begin{matrix}{T_{S} > R_{\text{PoI}} \cdot C_{\text{PoI}}} & \text{­­­(13)}\end{matrix}$

in further temperature and SoC (state of charge) regions

$\begin{matrix}\begin{array}{r}{\text{Δ}U_{\text{PoI}}(t) = \left( \frac{\partial U_{\text{PoI}}}{\partial t} \right|_{T_{S}}\left( {\text{Δ}t + \frac{\partial U_{\text{PoI}}}{\partial T} \cdot \frac{\partial T}{\partial t}} \right|_{T_{S}}\text{Δ}t + \ldots} \\{\left( {\cdots + \frac{\partial U_{\text{PoI}}}{\partial\text{SoC}} \cdot \frac{\partial\text{SoC}}{\partial t}} \right|_{T_{S}}\text{Δ}t + \cdots \neq 0}\end{array} & \text{­­­(14)}\end{matrix}$

cannot be negated. The change of the stochastic measuring noise cannotbe negated. The entire difference of the delayed system response leadsto results

$\begin{matrix}{\left( {\text{Δ}F(t) = \text{Δ}U_{\text{PoI}}(t)} \right|_{T_{S} > R_{\text{Diff}} \cdot C_{\text{PoI}}} + \text{Δ}F_{\text{stoch}}.} & \text{­­­(15)}\end{matrix}$

via usage of the backwards differences of the discrete voltage signals.

$\begin{matrix}{\delta U(t) = U(t) - U\left( {t - T_{S}} \right) = \delta I(t) \cdot R_{\text{Ohm}}(t) + \delta F(t)} & \text{­­­(16)}\end{matrix}$

For an improved overview, the discrete signals are combined into avector

$\begin{matrix}{\delta\overset{\rightarrow}{U}(t) = \begin{bmatrix}{\delta U(t)} & {\delta U\left( {t + T_{S}} \right)} & \cdots & {\delta U\left( {t + n \cdot T_{S}} \right)}\end{bmatrix}^{\text{T}}} & \text{­­­(17)}\end{matrix}$

having the length of the sampling size n that in particular correspondsto the number of battery cells 2. The system equation can now be writtenin a vector notation

$\begin{matrix}{\delta\overset{\rightarrow}{U}(t) = R_{\text{Ohm}}(t) \cdot \delta\overset{\rightarrow}{I}(t) + \delta\overset{\rightarrow}{F}(t)} & \text{­­­(18)}\end{matrix}$

One possible way of determining the ohmic resistance is therefore toapply the covariance of the voltage vector to the current vector

$\begin{matrix}{\frac{\text{Cov}\left( {\delta\overset{\rightarrow}{U}(t),\delta\overset{\rightarrow}{I}(t)} \right)}{\mathbb{V}\left( {\delta I(t)} \right)} = R_{\text{Ohm}}(t) + \frac{\text{Cov}\left( {\delta\overset{\rightarrow}{F}(t),\delta\overset{\rightarrow}{I}(t)} \right)}{\mathbb{V}\left( {\delta\overset{\rightarrow}{I}(t)} \right)}} & \text{­­­(19)}\end{matrix}$

on the condition that the present current vector is constant, and thevariance of the present current vector is equal to zero. The second termof equation 19 can be approximated by means of the equations (9) - (15):

$\begin{matrix}{\frac{\text{Cov}\left( {\delta\overset{\rightarrow}{F}(t),\delta\overset{\rightarrow}{I}(t)} \right)}{\mathbb{V}\left( {\delta\overset{\rightarrow}{I}(t)} \right)} \approx \left( {R_{\text{PoI}}(t)} \right|_{T_{S} > R_{\text{PoI}} \cdot C_{\text{PoI}}}} & \text{­­­(20)}\end{matrix}$

The covariance of cell voltage and cell current can be approximated asfollows,

$\begin{matrix}{\frac{\text{Cov}\left( {\delta\overset{\rightarrow}{U}(t),\delta\overset{\rightarrow}{I}(t)} \right)}{\mathbb{V}\left( {\delta\overset{\rightarrow}{I}(t)} \right)} = \left( {R_{\text{Ohm}}(t) + R_{\text{PoI}}(t)} \right|_{T_{S} > R_{\text{PoI}} \cdot C_{\text{PoI}}} \equiv R_{I}(t)} & \text{­­­(21)}\end{matrix}$

which is defined as an internal resistance R_(I)(t) of the respectivebattery cell 2. Applying the regression method to real data requiresfilter techniques, as real profiles contain phases in which the currentis constant or zero. Using the least squares method, the correlationcoefficient for evaluating the quality of a regression is

$\begin{matrix}{G(t) = \frac{\text{Cov}\left( {\delta\overset{\rightarrow}{U}(t),\delta\overset{\rightarrow}{I}(t)} \right)}{\sqrt{\mathbb{V}\left( {\delta\overset{\rightarrow}{I}(t)} \right) \cdot \mathbb{V}\left( {\delta\overset{\rightarrow}{U}(t)} \right)}} = R_{I}(t) \cdot \sqrt{\frac{\mathbb{V}\left( {\delta\overset{\rightarrow}{I}(t)} \right)}{\mathbb{V}\left( {\delta\overset{\rightarrow}{U}(t)} \right)}}.} & \text{­­­(22)}\end{matrix}$

as a quality factor G on the basis of the database of each sample forevaluating the assessment. This application comprises two differenttechniques of the regression method. One method requires a data backupof the respective sample in order to analytically calculate thecovariance and the correlation coefficient. The other algorithmapproximates the covariance and the correlation coefficient whileavoiding the need for an interim storage. The two algorithms arecompletely described in the following, including filters used, startingvalues and definitions.

B. Summary of the Regression Method Having Data Backup (RM)

Notation and definitions:

-   sample vector x _(k) = [x_(k) x_(k+I) x_(k+2) ··· x_(k+nopt) ]^(T)-   for the current δI _(k) = I _(k) - I_(k-1)-   for the voltage δU _(k) = U _(k) - U _(k-1)-   having the optimised sample size n_(opt) = length (δI _(k)) = length    (δI _(k))

Initialization: k = 0

R_(I, 0)^(*) = 0

Calculation: k > 0

$R_{I,k} = \frac{\text{Cov}\left( {\delta{\overset{\rightarrow}{I}}_{k},\delta{\overset{\rightarrow}{U}}_{k}} \right)}{\mathbb{V}\left( {\delta{\overset{\rightarrow}{I}}_{k}} \right)}$

$G_{\mspace{6mu} k} = \frac{\text{Cov}\left( {\delta{\overset{\rightarrow}{I}}_{k},\delta{\overset{\rightarrow}{U}}_{k}} \right)}{\sqrt{\mathbb{V}\left( {\delta{\overset{\rightarrow}{I}}_{k}} \right) \cdot \mathbb{V}\left( {\delta{\overset{\rightarrow}{U}}_{k}} \right)}} = R_{I,k} \cdot \sqrt{\frac{\mathbb{V}\left( {\delta{\overset{\rightarrow}{I}}_{k}} \right)}{\mathbb{V}\left( {\delta{\overset{\rightarrow}{U}}_{k}} \right)}}$

$R_{I,k} = \left\{ \begin{array}{ll}{R_{I,k},} & {\text{if}\quad\mathbb{V}\left( {\delta{\overset{\rightarrow}{I}}_{k}} \right) \neq 0 \cap R_{I,k} \in {\mathbb{R}}^{+} \cap R_{I,k} < \varepsilon} \\{R_{I,k - 1},} & \text{else}\end{array} \right)$

R_(I, k)^(*) = (1 − τ) ⋅ R_(I, k − 1)^(*) + τ ⋅ R_(I, k)

C. Summary of the Recursive Regression Method (RRM)

Definitions:

$\delta{\overset{\rightarrow}{d}}_{k} = \begin{bmatrix}I_{k} & U_{k}\end{bmatrix}^{\text{T}} - \begin{bmatrix}I_{k - 1} & U_{k - 1}\end{bmatrix}^{\text{T}}$

Initialization: k = 0

R_(I, 0)^(*) = 0,

${\overset{\rightarrow}{\mu}}_{0} = \begin{bmatrix}0 & 0\end{bmatrix}^{\text{T}}.$

$C_{0} = {\overset{\rightarrow}{\mu}}_{0} \otimes {\overset{\rightarrow}{\mu}}_{0}.$

Calculation of the output state: 0 ≤ k < n_(opt)

$C_{k + 1} = \left( {1 - \lambda} \right) \cdot \left( {C_{k} + \lambda \cdot \left( {\delta{\overset{\rightarrow}{d}}_{k + 1} - {\overset{\rightarrow}{\mu}}_{k}} \right) \otimes \left( {\delta{\overset{\rightarrow}{d}}_{k + 1} - {\overset{\rightarrow}{\mu}}_{k}} \right)} \right).$

${\overset{\rightarrow}{\mu}}_{k + 1} = \left( {1 - \lambda} \right) \cdot {\overset{\rightarrow}{\mu}}_{k} + \lambda \cdot \delta{\overset{\rightarrow}{d}}_{k + 1},$

Calculation for: k ≥ n_(opt)

$R_{I,k} = \frac{C_{k}\left( {1,2} \right)}{C_{k}\left( {1,1} \right)}$

$G_{\mspace{6mu} k} = \frac{C_{k}\left( {1,2} \right)}{\sqrt{C_{k}\left( {1,1} \right) \cdot C_{k}\left( {2,2} \right)}} = R_{I,k} \cdot \sqrt{\frac{C_{k}\left( {1,1} \right)}{C_{k}\left( {2,2} \right)}}$

$R_{I,k} = \left\{ \begin{array}{ll}{R_{I,k},} & {\text{if}\quad C_{k}\left( {1,1} \right) \neq 0 \cap R_{I,k} \in {\mathbb{R}}^{+}\mspace{6mu} \cap \mspace{6mu} R_{I,k} < \varepsilon} \\{R_{I,k - 1},} & \text{else}\end{array} \right)$

R_(I, k)^(*) = (1 − |τ)) ⋅ R_(I, k − 1)^(*) + τ ⋅ R_(I, k)

C_(k + 1) = (1 − λ) ⋅ (C_(k) + λ ⋅ (δd̃_(k + 1) − μ̃_(k)) ⊗ (δd̃_(k + 1) − μ̃_(k))).

μ̃_(k + 1) = (1 − λ) ⋅ μ̃_(k) + λ ⋅ δd̃_(k + 1).

D. Degradation (SoH) and Temperature Estimation

The output of the described data-based algorithms leads to a resistancedistribution R_(I,k) of the lithium-ion cells within the vehicle battery1. This distribution is comprised of degradation (SoH) and temperaturegradients, and tolerances in cell production and module design. In orderto differentiate between these origins, a reference distribution

$\begin{matrix}{\overset{\rightarrow}{N_{Rdj}} = {\overset{\rightarrow}{R_{I,k,j\cdot}}/\overline{\overline{R_{I,k,j}}}}} & \text{­­­(23)}\end{matrix}$

is offset in a relaxed state. The index j represents the respectivecycle. Short-term changes of this reference distribution

$\begin{matrix}{\overset{\rightarrow}{RR} = {{{\overset{\rightarrow}{N}}_{j}.}/\overset{\rightarrow}{N_{Rxi,j}}}} & \text{­­­(24)}\end{matrix}$

can be based on temperature gradients within the battery. Long-termshifts between the following reference distributions

$\overset{\rightarrow}{RR} = {\overset{\rightarrow}{N_{{Re}\text{t}\, j\cdot}}/\overset{\rightarrow}{N_{{Re}tj-1}}}$

describe a rate of aging gradient that has an influence on thedegradation or the SoH.

List of reference characters 1 vehicle battery 2 battery cell 3 group 4temperature measuring unit 5 ports 6 battery housing 7 measuring unit 9control device 11 first battery cell 12 second battery cell 15distribution function

1-10. (canceled)
 11. A method for determining a respective temperatureof a plurality of battery cells (2) of a vehicle battery (1), comprisingthe steps of: determination of (S1) measured values comprising a firstvoltage of at least one first battery cell (11), a second voltage of atleast one second battery cell (12), and at least one respective currentthat flows through the first and second battery cells (11, 12);determination of (S2) a first measured resistance of the first batterycell (11) and a second measured resistance of the second battery cell(12) from the measured values; determination (S4) of a referenceresistance; determination (S5) of a first resistance relationship fromthe first measured resistance and the reference resistance and of asecond resistance relationship from the second measured resistance andthe reference resistance; determination (S6) of a measured temperatureof the first battery cell (11); and determination (S7) of a computedtemperature of the second battery cell (12) based on the measuredtemperature of the first battery cell (11), the first resistancerelationship, and the second resistance relationship according to apredetermined requirement.
 12. The method according to claim 11, whereinthe reference resistance is an average value of the first measuredresistance and the second measured resistance.
 13. The method accordingto claim 11, wherein the predetermined requirement contains adistribution function (19) and/or a value table that assigns exactly onerespective value for the computed temperature to a plurality of valuesfor the second resistance relationship depending on the measuredtemperature and the first resistance relationship.
 14. The methodaccording to claim 13, wherein the distribution function (19) and/or thevalue table is first normalized depending on the first resistancerelationship and the measured temperature and the computed temperatureis then derived from the normalized distribution function (19) and/orthe value table depending on the second resistance relationship.
 15. Themethod according to claim 11, further comprising the step of determininga respective second resistance relationship for a plurality of secondbattery cells (12), wherein from the respective second resistancerelationship, the predetermined requirement, the measured temperature,and the first resistance relationship, a respective value for therespective computed temperature of each of the plurality of secondbattery cell (12) is determined.
 16. The method according to claim 11,further comprising determining a respective measured temperature and arespective first resistance relationship for a plurality of firstbattery cells (11), wherein from the respective measured temperature andthe respective first resistance relationship for the plurality of firstbattery cells (11), the predetermined requirement, and the secondresistance relationship, a value for the computed temperature of thesecond battery cell (12) is determined.
 17. The method according toclaim 16, wherein at least one measured temperature of one of theplurality of first battery cells (11) disposed on an edge of the vehiclebattery (1) and at least one measured temperature of one of theplurality of first battery cells (11) disposed in a center of thevehicle battery (1) is determined.
 18. The method according to claim 11,further comprising the step of checking (S3) whether a reference stateis present, wherein a correlation coefficient G is greater than adetermined value and/or respective measured temperatures of a pluralityof first battery cells do not exceed a determined measure in thereference state.
 19. An apparatus for determining a respectivetemperature of a plurality of battery cells (2) of a vehicle battery(1), comprising: a control device (9), wherein the control device isconfigured to: receive measured values comprising a first voltage of atleast one first battery cell (11), a second voltage of at least onesecond battery cell (12), and at least one respective current that flowsthrough the first and second battery cells (11, 12); determine a firstmeasured resistance of the first battery cell (11) and a second measuredresistance of the second battery cell (12) from the measured values;determine a reference resistance; determine a first resistancerelationship from the first measured resistance and the referenceresistance and a second resistance relationship from the second measuredresistance and the reference resistance; receive a measured temperatureof the first battery cell (11); and determine a computed temperature ofthe second battery cell (12) based on the measured temperature of thefirst battery cell (11), the first resistance relationship, and thesecond resistance relationship according to a predetermined requirement.20. A vehicle battery (1), comprising: the control device (9) accordingto claim 19; at least one first battery cell (11) and a respectivetemperature measuring unit for determining a respective measuredtemperature of the at least one battery cell (11); at least one secondbattery cell (12); and a measuring device for determining measuredvalues comprising a first voltage of the least one first battery cell(11), a second voltage of the at least one second battery cell (12), andat least one respective current that flows through the first and secondbattery cells (11, 12).